The new sign for a local business contains two different sections. One of the sections will be produced from wood, while the other will be metal. Metal is three times as expensive as wood. The cost of metal needed for each sign is proportional to the diameter of the sign, while the cost of wood needed is proportional to the square of the diameter. If the diameter of the sign is doubled, then the total cost of the materials will be tripled. What percentage (to the nearest 1%) of the sign is metal?

Answer:

24) A

The explanation to this question is a bit complicated (or confusing). If you have any question, feel free to comment down in the comments section.

It is important to use variables (such as ‘x’ or ‘y’ or ‘z’) when we are not given any values in the question.

Let cost of wood be = W
We know cost of metal is three times cost of wood.
Hence, cost of metal = 3W

Let amount of wood used = r
Therefore, total cost of wood = (W x t)

Let amount of metal used = u
Therefore, total cost of metal = (3W x u)

Total cost of sign = Wr + 3Wu

Let the diameter of sign = D
Wood is proportional to size of diameter: r = D²
Metal is proportional to the square of diameter: u = D

Since the diameter is doubled (2D), we get the following equations:

r = (2D)² = 4D² = 4r (Since D² = r)
u = 2D = 2u (Since D = u)

New total cost = (W x 4r) + (3W x 2u) = 4Wr + 6Wu

Since the diameter is doubled, the total cost will be tripled.

Therefore the new cost is three times the old cost.

4Wr + 6Wu = 3[Wr + 3Wu]
4Wr + 6Wu = 3Wr + 9Wu
Wr = 3Wu
r = 3u

Percentage of metal = Amount of metal ÷ Total amount of material
Total amount of material = amount of wood (r) + amount of metal (u)
Amount of metal = u

Percentage of metal = u ÷ (r + u)
Since r = 3u, we get:
Percentage of metal = u ÷ (3u + u) = u ÷ 4u = 1/4 = 25%